There is an issue here that we have been talking around and around
without ever connecting up on, namely
Has McLarty defined 'morphism' or 'category'?
Here I'm thinking of a theory as serving to characterize the
individuals it is a theory of, as opposed to characterizing the
universe of those individuals itself. This is in agreement with the
idea that ZF defines the notion of "set."
Neither Sol nor (I think) Torkel have addressed the point I drew
attention to on Tuesday:
>From: "Vaughan R. Pratt" <pratt at CS.Stanford.EDU>
>Date: Tue, 18 Nov 1997 19:46:36 -0800
>Colin is exactly right here, but it is easy to miss an absolutely
>crucial point that he makes towards the end of his message, without
>which it is not at all apparent what the problem is with Sol's
>argument.
>>One could easily suppose Sol and Colin were talking at crosspurposes.
>For Sol, set theory is the theory of the sets forming the ZF
>universe---the individuals in the theory are sets. For Colin, category
>theory is the theory of the objects forming a single category---the
>individuals in the theory are objects.
Sol says,
>From: Solomon Feferman <sf at Csli.Stanford.EDU>
>Date: Wed, 19 Nov 1997 23:46:02 -0800 (PST)
>McLarty wants to say in place of "What is a category?"--"What are the
>first order axioms of categories?" and likewise for sets, etc., thus
>seemingly putting everything on a par: no priorities at all, and no need
>to posit collections and operations. Similarly, supposedly, for groups,
>rings, topological spaces, etc. But if we want to be able to say: the
>set of all permutations of a set of n elements forms a group under
>composition, where are we? This is a specific structure. What are its
>elements? What is the operation?
I answered this on Tuesday by pointing out that after defining
"category" Colin defines "category of categories". This is the level
at which one reasons externally to categories, as requested here by
Sol.
What I'm not understanding is why Colin himself is not drawing more
attention to this question, as it seems to be one of the larger
obstacles to any meeting of the minds here. Instead of:
>Date: Wed, 19 Nov 1997 13:28:26 -0500 (EST)
>This claim seems quite groundless to me. It seems to be refuted by many
>widely published examples, some of which I mentioned in my post. In
>fact, it seems to me SO ENTIRELY wrong that I think I may misunderstand
>it in some fundamental way. I would like to see a defense of this
>claim.
what would get at the difference more directly would be to address
Sol's (and Torkel's?) specific concern about the axioms seeming to be
useful only for axiomatizing a single category, as opposed to category
theory.
Colin has already agreed that what I said in that connection is
accurate, so at this point it should just be a matter of his backing me
up a second time and agreeing that this is indeed what he intended to
convey.
Vaughan Pratt